On sets of uniqueness for harmonic functions in the unit circle

Yu. Ya. Vymenets

Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 23, Tome 222 (1995), pp. 78-123

Résumé

The results of this paper show that the structure of sets mentioned in the title is not trivial. For example, it is shown that there exist countable sets of uniqueness for logarithmic potential, i.e., closed countable subsets $E$ of the unit circle $\mathbb T$ such that $$ f\in C(\mathbb T),\ f\mid_E=0,\ U^f\mid_E=0\ \Rightarrow f\equiv0. $$ Here $U^f(z)=\frac1\pi\int_0^{2\pi}f(e^{i\theta})\log\frac1{|z-e^{i\theta}|}\,d\theta$. On the other hand, it is shoum that every countable porous closed subset of $\mathbb T$ is a nonuniqueness set. Bibliography: 9 titles.